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6.254 : Game Theory with Engineering Applications
Lecture 9: Computation of NE in ﬁnite games
Asu Ozdaglar
MIT
March 4, 2010
1
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View Full Document Game Theory: Lecture 9
Introduction
Introduction
In this lecture, we study various approaches for the computation of
mixed Nash equilibrium for ﬁnite games.
Our focus will mainly be on two player ﬁnite games (i.e., bimatrix
games).
We will also mention extensions to games with multiple players and
continuous strategy spaces at the end.
The two survey papers [von Stengel 02] and [McKelvey and
McLennan 96] provide good references for this topic.
2
Game Theory: Lecture 9
ZeroSum Finite Games
ZeroSum Finite Games
We consider a zerosum game where we have two players. Assume that
player 1 has
n
actions and player 2 has
m
actions.
We denote the
n
×
m
payoﬀ matrices of player 1 and 2 by
A
and
B
.
Let
x
denote the mixed strategy of player 1, i.e.,
x
∈
X
, where
X
=
{
x

n
∑
i
=
1
x
i
=
1,
x
i
≥
0
}
,
and
y
denote the mixed strategy of player 2, i.e.,
y
∈
Y
, where
Y
=
{
y

m
∑
j
=
1
y
j
=
1,
y
j
≥
0
}
.
Given a mixed strategy proﬁle
(
x
,
y
)
, the payoﬀs of player 1 and player 2
can be expressed in terms of the payoﬀ matrices as,
u
1
(
x
,
y
) =
x
T
Ay
,
u
2
(
x
,
y
) =
x
T
By
.
3
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View Full Document Game Theory: Lecture 9
ZeroSum Finite Games
ZeroSum Finite Games
Recall the deﬁnition of a Nash equilibrium: A mixed strategy proﬁle
(
x
*
,
y
*
)
is a mixed strategy Nash equilibrium if and only if
(
x
*
)
T
Ay
*
≥
x
T
Ay
*
,
for all
x
∈
X
,
(
x
*
)
T
By
*
≥
(
x
*
)
T
By
,
for all
y
∈
Y
.
For zerosum games, we have
B
=

A
, hence the preceding relation
becomes
(
x
*
)
T
Ay
*
≤
(
x
*
)
T
Ay
,
for all
y
∈
Y
.
Combining the preceding, we obtain
x
T
Ay
*
≤
(
x
*
)
T
Ay
*
≤
(
x
*
)
T
AY
,
for all
x
∈
X
,
y
∈
Y
,
i.e.,
(
x
*
,
y
*
)
is a
saddle point of the function
x
T
Ay
deﬁned over
X
×
Y
.
Note that a vector
(
x
*
,
y
*
)
is a saddle point if
x
*
∈
X
,
y
*
∈
Y
, and
sup
x
∈
X
x
T
Ay
*
= (
x
*
)
T
Ay
*
=
inf
y
∈
Y
(
x
*
)
T
Ay
.
(1)
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Game Theory: Lecture 9
ZeroSum Finite Games
ZeroSum Finite Games
For any function
φ
:
X
×
Y
7→
R
, we have the
minimax inequality
:
sup
x
∈
X
inf
y
∈
Y
φ
(
x
,
y
)
≤
inf
y
∈
Y
sup
x
∈
X
φ
(
x
,
y
)
,
(2)
Proof:
To see this, for every
¯
x
∈
X
, write
inf
y
∈
Y
φ
(
¯
x
,
y
)
≤
inf
y
∈
Y
sup
x
∈
X
φ
(
x
,
y
)
and take the supremum over
¯
x
∈
X
of the lefthand side.
Eq. (1) implies that
inf
y
∈
Y
sup
x
∈
X
x
T
Ay
≤
sup
x
∈
X
x
T
Ay
*
= (
x
*
)
T
Ay
*
=
inf
y
∈
Y
(
x
*
)
T
Ay
≤
sup
x
∈
X
inf
y
∈
Y
x
T
Ay
,
which combined with the minimax inequality [cf. Eq. (2)], proves that
equality holds throughout in the preceding.
Hence, a mixed strategy proﬁle
(
x
*
,
y
*
)
is a Nash equilibrium if and only if
(
x
*
)
T
Ay
*
=
inf
y
∈
Y
sup
x
∈
X
x
T
Ay
=
sup
x
∈
X
inf
y
∈
Y
x
T
Ay
.
We refer to
(
x
*
)
T
Ay
*
as the
value of the game
.
5
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View Full Document Game Theory: Lecture 9
ZeroSum Finite Games
ZeroSum Finite Games
We next show that ﬁnding the mixed strategy Nash equilibrium strategies
and the value of the game can be written as a pair of linear optimization
problems.
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This note was uploaded on 05/08/2010 for the course CS 6.254 taught by Professor Asuozdaglar during the Spring '10 term at MIT.
 Spring '10
 AsuOzdaglar

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